Simulating the Hydrologic Simulating the Hydrologic ResponseResponse

“Moving Water to the Basin Outlet”

Presented By

Dennis Johnson

For

COMET Hydrometeorology 99-2

Friday, 25 June 1999

Unit Hydrograph TheoryUnit Hydrograph Theory

• Sherman - 1932

• Horton - 1933

• Wisler & Brater - 1949 - “the hydrograph of surface runoff resulting from a relatively short, intense rain, called a unit storm.”

• The runoff hydrograph may be “made up” of runoff that is generated as flow through the soil (Black, 1990).

Unit Hydrograph “Lingo”Unit Hydrograph “Lingo”

• Duration

• Lag Time

• Time of Concentration

• Rising Limb

• Recession Limb (falling limb)

• Peak Flow

• Time to Peak (rise time)

• Recession Curve

• Separation

• Base flow

Graphical RepresentationGraphical Representation

Lag time

Time of concentration

Duration of excess precipitation.

Base flow

Methods of Developing UHG’sMethods of Developing UHG’s

• From Streamflow Data

• Synthetically– Snyder– SCS– Time-Area (Clark, 1945)

• “Fitted” Distributions

Unit HydrographUnit Hydrograph

• The hydrograph that results from 1-inch of excess precipitation (or runoff) spread uniformly in space and time over a watershed for a given duration.

• The key points :1-inch of EXCESS precipitationSpread uniformly over space - evenly over the watershedUniformly in time - the excess rate is constant over the time

intervalThere is a given duration

Derived Unit HydrographDerived Unit HydrographStreamflow & Assumed Baseflow

0

100

200

300

400

500

600

700

800

0 5 10 15 20 25 30

Time (hrs)

cfs

Derived Unit HydrographDerived Unit Hydrograph

Rules of Thumb :… the storm should be fairly uniform in nature and the excess precipitation should be equally as uniform throughout the basin. This may require the initial conditions throughout the basin to be spatially similar. … Second, the storm should be relatively constant in time, meaning that there should be no breaks or periods of no precipitation. … Finally, the storm should produce at least an inch of excess precipitation (the area under the hydrograph after

correcting for baseflow).

Deriving a UHG from a StormDeriving a UHG from a Stormsample watershed = 5.4 mi2sample watershed = 5.4 mi2

0

100

200

300

400

500

600

700

800

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Separation of BaseflowSeparation of Baseflow

... generally accepted that the inflection point on the recession limb of a hydrograph is the result of a change in the controlling physical processes of the excess precipitation flowing to the basin outlet.

In this example, baseflow is considered to be a straight line connecting that point at which the hydrograph begins to rise rapidly and the inflection point on the recession side of the hydrograph.

the inflection point may be found by plotting the hydrograph in semi-log fashion with flow being plotted on the log scale and noting the time at which the recession side fits a straight line.

Semi-log PlotSemi-log Plot

1

10

100

1000

0 5 10 15 20 25 30

Assuming inflection @ ~ hour 17 (160 cfs)

Hydrograph & BaseflowHydrograph & Baseflow

0

100

200

300

400

500

600

700

800

0 5 10 15 20 25 30

Separate BaseflowSeparate Baseflow

-100

0

100

200

300

400

500

600

700

800

0 5 10 15 20 25 30

Represent Surface Flow as Discrete Blocks Represent Surface Flow as Discrete Blocks (by averaging)(by averaging)

-100

0

100

200

300

400

500

600

700

Your Spreadsheet :Your Spreadsheet :

Time Flow BaseflowSurface Runoff

Average Flow

Cubic Feet of Runoff Precipitation

0 32 32 0 0 0 0.51 33 33 0 0 0 1.22 32 32 0 0 0 1.83 32 32 0 23.9286 86142.86 0.34 89 41.142857 47.85714 110.286 397028.6 0.65 223 50.285714 172.7143 281.643 1013914 0.16 450 59.428571 390.5714 471 16956007 620 68.571429 551.4286 576.857 20766868 680 77.714286 602.2857 603.214 21715719 691 86.857143 604.1429 596.571 2147657

10 685 96 589 561.929 202294311 640 105.14286 534.8571 485.286 174702912 550 114.28571 435.7143 388.643 139911413 465 123.42857 341.5714 289.5 104220014 370 132.57143 237.4286 192.857 694285.715 290 141.71429 148.2857 103.714 373371.416 210 150.85714 59.14286 29.5714 106457.117 160 160 0 0 018 123 123 0 0 019 95 95 0 0 020 77 77 0 0 021 58 58 0 0 022 46 46 0 0 023 39 39 0 0 024 33 33 0 0 0

16974000

Sample CalculationsSample Calculations• The total volume of runoff was calculated as

16,974,000 cubic feet.

• The basin average would now be :

inchesarea

volumerunoff 35.112*

)43560*640*4.5(

000,974,16

Obtain UHG OrdinatesObtain UHG Ordinates

• The ordinates of the unit hydrograph are obtained by dividing each flow in the direct runoff hydrograph by the depth of excess precipitation.

• In this example, the units of the unit hydrograph would be cfs/inch (of excess precipitation).

Final UHGFinal UHG

-100

0

100

200

300

400

500

600

700

800

0 5 10 15 20 25 30

Flow

Surface Runoff

UHG

Determine Duration of UHGDetermine Duration of UHG• The duration of the derived unit hydrograph is found by examining the

precipitation for the event and determining that precipitation which is in excess.

• This is generally accomplished by plotting the precipitation in hyetograph form and drawing a horizontal line such that the precipitation above this line is equal to the depth of excess precipitation as previously determined.

• This horizontal line is generally referred to as the -index and is based on the assumption of a constant or uniform infiltration rate.

• The uniform infiltration necessary to cause 1.35 inches of excess precipitation was determined to be approximately 0.82 inches per hour.

Estimating Excess Precip.Estimating Excess Precip.

0

0.2

0.40.6

0.8

1

1.2

1.41.6

1.8

2

1 2 3 4 5 6

Changing the DurationChanging the Duration• Very often, it will be necessary to change the duration of the unit

hydrograph.

• If unit hydrographs are to be averaged, then they must be of the same duration.

• Also, convolution of the unit hydrograph with a precipitation event requires that the duration of the unit hydrograph be equal to the time step of the incremental precipitation.

• The most common method of altering the duration of a unit hydrograph is by the S-curve method.

• The S-curve method involves continually lagging a unit hydrograph by its duration and adding the ordinates.

• For the present example, the 6-hour unit hydrograph is continually lagged by 6 hours and the ordinates are added.

Develop S-CurveDevelop S-Curve

-500

0

500

1000

1500

2000

2500

3000

3500

4000

0 10 20 30 40 50

Convert to 1-Hour DurationConvert to 1-Hour Duration

• To arrive at a 1-hour unit hydrograph, the S-curve is lagged by 1 hour and the difference between the two lagged S-curves is found to be a 1 hour unit hydrograph.

• However, because the S-curve was formulated from unit hydrographs having a 2 hour duration of uniformly distributed precipitation, the hydrograph resulting from the subtracting the two S-curves will be the result of 1/2 of an inch of precipitation.

• Thus the ordinates of the newly created 1-hour unit hydrograph must be multiplied by 2 in order to be a true unit hydrograph.

• The 1-hour unit hydrograph should have a higher peak which occurs earlier than the 2-hour unit hydrograph.

Final 1-hour UHGFinal 1-hour UHG

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14 16

1, 2, 6, &12 Hr UHG’s1, 2, 6, &12 Hr UHG’s

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30

Average Several UHG’sAverage Several UHG’s• It is recommend that several unit hydrographs be derived and averaged.

• The unit hydrographs must be of the same duration in order to be properly averaged.

• It is often not sufficient to simply average the ordinates of the unit hydrographs in order to obtain the final unit hydrograph. A numerical average of several unit hydrographs which are different “shapes” may result in an “unrepresentative” unit hydrograph.

• It is often recommended to plot the unit hydrographs that are to be averaged. Then an average or representative unit hydrograph should be sketched or fitted to the plotted unit hydrographs.

• Finally, the average unit hydrograph must have a volume of 1 inch of runoff for the basin.

Synthetic UHG’sSynthetic UHG’s

• Snyder

• SCS

• Time-area

• IHABBS Implementation Plan :

NOHRSC Homepagehttp://www.nohrsc.nws.gov/

http://www.nohrsc.nws.gov/98/html/uhg/index.html

SnyderSnyder

• Since peak flow and time of peak flow are two of the most important parameters characterizing a unit hydrograph, the Snyder method employs factors defining these parameters, which are then used in the synthesis of the unit graph (Snyder, 1938).

• The parameters are Cp, the peak flow factor, and Ct, the lag factor.

• The basic assumption in this method is that basins which have similar physiographic characteristics are located in the same area will have similar values of Ct and Cp.

• Therefore, for ungaged basins, it is preferred that the basin be near or

similar to gaged basins for which these coefficients can be determined.

Basic RelationshipsBasic Relationships3.0)( catLAG LLCt

5.5LAG

durationtt

)(25.0 .. durationdurationaltLAGlagalt tttt

83 LAG

baset

t

LAG

ppeak t

ACq

640

Final ShapeFinal ShapeThe final shape of the Snyder unit hydrograph is controlled by the

equations for width at 50% and 75% of the peak of the UHG:

SCSSCS

SCS Dimensionless UHG Features

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

T/Tpeak

Q/Q

pe

ak

Flow ratios

Cum. Mass

Dimensionless RatiosDimensionless RatiosTime Ratios

(t/tp)Discharge Ratios

(q/qp)Mass Curve Ratios

(Qa/Q)0 .000 .000.1 .030 .001.2 .100 .006.3 .190 .012.4 .310 .035.5 .470 .065.6 .660 .107.7 .820 .163.8 .930 .228.9 .990 .300

1.0 1.000 .3751.1 .990 .4501.2 .930 .5221.3 .860 .5891.4 .780 .6501.5 .680 .7001.6 .560 .7511.7 .460 .7901.8 .390 .8221.9 .330 .8492.0 .280 .8712.2 .207 .9082.4 .147 .9342.6 .107 .9532.8 .077 .9673.0 .055 .9773.2 .040 .9843.4 .029 .9893.6 .021 .9933.8 .015 .9954.0 .011 .9974.5 .005 .9995.0 .000 1.000

Triangular RepresentationTriangular RepresentationSCS Dimensionless UHG & Triangular Representation

0

0.2

0.4

0.6

0.8

1

1.2

0.0 1.0 2.0 3.0 4.0 5.0

T/Tpeak

Q/Q

pea

k

Flow ratios

Cum. Mass

Triangular

Excess Precipitation

D

Tlag

Tc

TpTb

Point of Inflection

Triangular RepresentationTriangular Representationpb T x 2.67 T

ppbr T x 1.67 T - T T

)T + T( 2

q =

2

Tq +

2

Tq = Q rp

prppp

T + T

2Q = q

rpp

T + T

Q x A x 2 x 654.33 = q

rpp The 645.33 is the conversion used for

delivering 1-inch of runoff (the area under the unit hydrograph) from 1-square

mile in 1-hour (3600 seconds). T

Q A 484 = q

pp

SCS Dimensionless UHG & Triangular Representation

0

0.2

0.4

0.6

0.8

1

1.2

0.0 1.0 2.0 3.0 4.0 5.0

T/Tpeak

Q/Q

pea

k

Flow ratios

Cum. Mass

Triangular

Excess Precipitation

D

Tlag

Tc

TpTb

Point of Inflection

484 ?484 ?

Comes from the initial assumption that 3/8 of the volume under the UHG is under the rising limb and the remaining 5/8

is under the recession limb.

General Description Peaking Factor Limb Ratio(Recession to Rising)

Urban areas; steep slopes 575 1.25Typical SCS 484 1.67

Mixed urban/rural 400 2.25Rural, rolling hills 300 3.33Rural, slight slopes 200 5.5

Rural, very flat 100 12.0

T

Q A 484 = q

pp

Duration & Timing?Duration & Timing?

L + 2

D = T p

cTL *6.0

L = Lag time

pT 1.7 D Tc

T = T 0.6 + 2

Dpc

For estimation purposes : cT 0.133 D

Again from the triangle

Time of ConcentrationTime of Concentration

• Regression Eqs.

• Segmental Approach

A Regression EquationA Regression Equation

TlagL S

Slope

08 1 0 7

1900 05

. ( ) .

(% ) .

where : Tlag = lag time in hoursL = Length of the longest drainage path in feetS = (1000/CN) - 10 (CN=curve number)%Slope = The average watershed slope in %

Segmental ApproachSegmental Approach

• More “hydraulic” in nature

• The parameter being estimated is essentially the time of concentration or longest travel time within the basin.

• In general, the longest travel time corresponds to the longest drainage path

• The flow path is broken into segments with the flow in each segment being represented by some type of flow regime.

• The most common flow representations are overland, sheet, rill and

gully, and channel flow.

A Basic ApproachA Basic Approach 2

1

kSV

McCuen (1989) and SCS (1972) provide values of k for several flow situations

(slope in %)

K Land Use / Flow Regime0.25 Forest with heavy ground litter, hay meadow (overland flow)0.5 Trash fallow or minimum tillage cultivation; contour or strip

cropped; woodland (overland flow)0.7 Short grass pasture (overland flow)0.9 Cultivated straight row (overland flow)1.0 Nearly bare and untilled (overland flow); alluvial fans in

western mountain regions1.5 Grassed waterway2.0 Paved area (sheet flow); small upland gullies

Flow Type KSmall Tributary - Permanent or intermittent

streams which appear as solid or dashedblue lines on USGS topographic maps.

2.1

Waterway - Any overland flow route whichis a well defined swale by elevation

contours, but is not a stream section asdefined above.

1.2

Sheet Flow - Any other overland flow pathwhich does not conform to the definition of

a waterway.

0.48

Sorell & Hamilton, 1991

Triangular ShapeTriangular Shape

• In general, it can be said that the triangular version will not cause or introduce noticeable differences in the simulation of a storm event, particularly when one is concerned with the peak flow.

• For long term simulations, the triangular unit hydrograph does have a potential impact, due to the shape of the recession limb.

• The U.S. Army Corps of Engineers (HEC 1990) fits a Clark unit hydrograph to match the peak flows estimated by the Snyder unit hydrograph procedure.

• It is also possible to fit a synthetic or mathematical function to the peak flow and timing parameters of the desired unit hydrograph.

• Aron and White (1982) fitted a gamma probability distribution using peak flow and time to peak data.

FittingFitting aa GammaGamma DistributionDistribution

)1(),;(

1

ab

etbatf

a

bta

0.0000

50.0000

100.0000

150.0000

200.0000

250.0000

300.0000

350.0000

400.0000

450.0000

500.0000

0.0000 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000

Time-AreaTime-Area

Time-AreaTime-Area

Time

Q % Area

Time

100%

Timeof conc.

Time-AreaTime-Area

Hypothetical ExampleHypothetical Example

• A 190 mi2 watershed is divided into 8 isochrones of travel time.

• The linear reservoir routing coefficient, R, estimated as 5.5 hours.

• A time interval of 2.0 hours will be used for the computations.

WatershedBoundary

Isochrones

2

345

66

7

8

6

6

5

7

7

1

0

Rule of ThumbRule of Thumb

R - The linear reservoir routing coefficient can be estimated as approximately 0.75

times the time of concentration.

Basin BreakdownBasin Breakdown

MapArea #

BoundingIsochrones

Area(mi2)

CumulativeArea (mi2)

CumulativeTime (hrs)

1 0-1 5 5 1.02 1-2 9 14 2.03 2-3 23 37 3.04 3-4 19 58 4.05 4-5 27 85 5.06 5-6 26 111 6.07 6-7 39 150 7.08 7-8 40 190 8.0

TOTAL 190 190 8.0

WatershedBoundary

Isochrones

2

345

66

7

8

6

6

5

7

7

1

0

Incremental AreaIncremental Area

0

5

10

15

20

25

30

35

40

Incr

emen

tal

Are

a (s

qau

re m

iles

)

1 2 3 4 5 6 7 8

Time Increment (hrs)

WatershedBoundary

Isochrones

2

345

66

7

8

6

6

5

7

7

1

0

Cumulative Time-Area CurveCumulative Time-Area Curve

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140 160 180 200

Time (hrs)

Cu

mu

lati

ve A

rea

(sq

aure

mil

es)

WatershedBoundary

Isochrones

2

345

66

7

8

6

6

5

7

7

1

0

Trouble Getting a Time-Area Curve?Trouble Getting a Time-Area Curve?

0.5) Ti (0for 414.1 5.1 ii TTA

1.0) Ti (0.5for )1(414.11 5.1 ii TTA

Synthetic time-area curve - The U.S. Army Corps of Engineers (HEC 1990)

Instantaneous UHGInstantaneous UHG)1(

)1( iii

IUHccIIUH

tR

tc

2

2

t = the time step used n the calculation of the translation unit hydrograph

The final unit hydrograph may be found by averaging 2 instantaneous unit hydrographs that are a t time step apart.

ComputationsComputationsTime(hrs)

(1)

Inc.Area(mi2)(2)

Inc.TranslatedFlow (cfs)

(3)

Inst. UHG

(4)

IUHGLagged 2

hours(5)

2-hrUHG(cfs)(6)

0 0 0 0 02 14 4,515 1391 0 7004 44 14,190 5333 1,391 3,3606 53 17,093 8955 5,333 7,1508 79 25,478 14043 8,955 11,50010 0 0 9717 14,043 11,88012 6724 9,717 8,22014 4653 6,724 5,69016 3220 4,653 3,94018 2228 3,220 2,72020 1542 2,228 1,89022 1067 1,542 1,30024 738 1,067 90026 510 738 63028 352 510 43030 242 352 30032 168 242 20034 116 168 14036 81 116 10038 55 81 7040 39 55 5042 26 39 3044 19 26 2046 13 19 2048 13

)1()1(

iiiIUHccIIUH

tR

tc

2

2

Incremental AreasIncremental Areas

0

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10

Time Increments (2 hrs)

Are

a In

crem

ents

(sq

uar

e m

iles

)

Incremental FlowsIncremental Flows

0

5000

10000

15000

20000

25000

30000

1 2 3 4 5 6

Time Increments (2 hrs)

Tra

nsl

ated

Un

it H

ydro

gra

ph

Instantaneous UHGInstantaneous UHG

0

2000

4000

6000

8000

10000

12000

14000

16000

0 10 20 30 40 50 60

Time (hrs)

Flo

w (

cfs/

inch

)

Lag & AverageLag & Average

0

2000

4000

6000

8000

10000

12000

14000

16000

0 10 20 30 40 50 60

Time (hrs)

Flo

w (

cfs/

inch

)

ConvolutionConvolution

Current Modeling Techniques in NWSCurrent Modeling Techniques in NWS

• NWSRFS Models use spatially and temporally averaged quantities.

• NWSRFS models are parametric(API) and conceptual (SAC-SMA).

• Spatial & Temporal “lumping” may lead to model errors.

Distributed Models - Basic Concept(s)Distributed Models - Basic Concept(s)

• Break basin into smaller “pieces” to lessen the effect of averaging.

• Allow for the horizontal and vertical interaction of water movement throughout the basin (horizontal is the greatest challenge).

Distributed Modeling AdvantagesDistributed Modeling Advantages

• The greatest advantage of distributed modeling is the ability to overcome the “loss” of data due to spatial and temporal averaging.

Distributed Modeling DISAdvantage(s)Distributed Modeling DISAdvantage(s)

• Data requirements (disk space) - NOT

• Computational requirements - STILL?

• Parameterization

• Calibration

Development of a Hydrologic ModelDevelopment of a Hydrologic Modelwith bothwith both

Spatially Distributed and Physically Based Spatially Distributed and Physically Based Capabilities - Capabilities -

Could we use this time-area concept?Could we use this time-area concept?

How About a Distributed UHG?How About a Distributed UHG?

IHABBS Implementation Plan :NOHRSC Homepage

http://www.nohrsc.nws.gov/http://www.nohrsc.nws.gov/98/html/uhg/index.html

Each Grid Cell ContributesEach Grid Cell Contributes

TIME

Volume

Now Follow Time-Area MethodNow Follow Time-Area Method

How About a “Dynamic UHG”How About a “Dynamic UHG”

Only certain cells contribute - based on storm location - therefore only route the contributing cells using the time-area method.

What are the Challenges?What are the Challenges?

• How much runoff from each cell?

• Lumped calibration being distributed!

• How about that “reservoiring” function?

• See USACE : HEC-HMS

• Many other models - no offense intended!!